# An ODE for tensor - possibility of the equation together with the initial condition at $t=0$ deciding the solution for all $t>0$

Let $$A_{ijl}(t,x) : [0,\infty) \times \mathbb{R}^n \to [\mathbb{R}^n]^3$$ be a smooth tensor field. That is, $$i,j,l \in \{1,2,3, \cdots, n\}$$

Further assume that $$A_{ijl}(t,x)=A_{jil}(t,x)$$ for all $$(t,x) \in [0,\infty) \times \mathbb{R}^n$$.

Also, let $$0=\lambda_1< \lambda_2 < \cdots < \lambda_n$$ be some fixed numbers and finally suppose that $$A_{ijl}(t,x)$$ satisfies the following ODE: $$$$\partial_t \sum_{l=1}^n A_{ijl}(t,x)= -\sum_{l=1}^n \lambda^2_l A_{ijl}(t,x) \text{ with } \sum_{l=1}^n A_{ijl}(0,x)=\delta_{ij}$$$$

Then, at least when $$t=0$$, we have $$$$\partial_t \sum_{l=1}^n A_{ssl}(t,x)\mid_{t=0}=-\lambda^2_s$$$$ for all $$s=1,2, \cdots, n$$ while $$$$\partial_t \sum_{l=1}^n A_{ijl}(t,x)\mid_{t=0}=0$$$$ for $$i \neq j$$.

From such information, is it possible to conclude, together with the smoothness assumption, that $$$$\sum_{l=1}^n A_{ijl}(t,x)=e^{-\lambda_i^2 t} \text{ if } i=j \text{ and } 0 \text{ otherwise}$$$$ for all $$t \geq 0$$?

It seems quite plausible for me, but I cannot really justify my guess for extrapolation to $$t>0$$. Could anyone please help me?

Edit : I add one more condition that $$A_{ijl}(0,x)=\delta_{il}\delta_{jl}$$ so that the above explanation makes sense.

These equations are not sufficient to determine $$\sum_l A_{ijl}$$ for $$n>1$$.

Here is a counterexample of a solution of the problem in the OP which contradicts the conjectured solution:

set $$n=2$$, $$\lambda_1=0$$, $$\lambda_2=1$$; all elements of $$A_{ijl}$$ are identically zero, except

$$A_{111}=1-t-t^2/2,\;\;A_{112}=t,$$ $$A_{221}=t,\;\;A_{222}=-1+2e^{-t}.$$

More generally, you can take any $$F_{ij}(t)=\sum_l A_{ijl}(t)$$, and define $$A_{ijl}=-\left(\sum_{k=2}^n\lambda_k^{-2}\right)^{-1}F'_{ij},\;\;l\geq 2,$$ $$A_{ij1}=F_{ij}+(n-1)\left(\sum_{k=2}^n\lambda_k^{-2}\right)^{-1}F'_{ij}.$$

• I am only interested in the sum of $A$ over the index $l$. I don't need details about individual components of $A$. In that case, is my tentative solution on $\sum_l A_{ijl}$ correct? Commented Jul 2, 2023 at 19:12
• I added an explicit counterexample to my answer. Commented Jul 2, 2023 at 21:20
• I think there is no restriction at all on the solution; I added this to the answer. Commented Jul 3, 2023 at 6:25
• no, as I wrote, you can satisfy the ODE for any function $F_{ij}(t)=\sum_l A_{ijl}(t)$ ; just choose a bounded function. Commented Jul 3, 2023 at 11:01
• yes, any time dependence will do. Commented Jul 3, 2023 at 11:37